Theorem spw 2064

Description: Weak version of the specialization scheme sp 2219. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2219 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2219 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2170 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2219 are spfw 2063 (minimal distinct variable requirements), spnfw 2009 (when 𝑥 is not free in ¬ 𝜑), spvw 2011 (when 𝑥 does not appear in 𝜑), sptruw 1836 (when 𝜑 is true), spfalw 2010 (when 𝜑 is false), and spvv 2018 (where 𝜑 is changed into 𝜓). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)

Hypothesis

Ref	Expression
spw.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spw	⊢ (∀𝑥𝜑 → 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem spw

Step	Hyp	Ref	Expression
1		ax-5 1940	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		ax-5 1940	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑)
3		ax-5 1940	. 2 ⊢ (¬ 𝜑 → ∀𝑦 ¬ 𝜑)
4		spw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5	1, 2, 3, 4	spfw 2063	1 ⊢ (∀𝑥𝜑 → 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810

This theorem is referenced by:  hba1w  2079  19.8aw  2082  exexw  2083  spaev  2084  ax12w  2168  rspw  3242  reldisj  4413  ralidmw  4477  dtruALT2  5341  bj-ssblem1  37296  bj-ax12w  37320  eu6w  43428